IEEE Transactions on Knowledge and Data Engineering（2023）
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摘要
$K$
-clique counting is a fundamental problem in network analysis which has attracted much attention in recent years. Computing the count of
$k$
-cliques in a graph for a large
$k$
(e.g.,
$k=8$
) is often intractable as the number of
$k$
-cliques increases exponentially w.r.t. (with respect to)
$k$
. Existing exact
$k$
-clique counting algorithms are often hard to handle large dense graphs, while sampling-based solutions either require a huge number of samples or consume very high storage space to achieve a satisfactory accuracy. To overcome these limitations, we propose a new framework to estimate the number of
$k$
-cliques which integrates both the exact
$k$
-clique counting technique and three novel color-based sampling techniques. The key insight of our framework is that we only apply the exact algorithm to compute the
$k$
-clique counts in the
sparse regions
of a graph, and use the proposed color-based sampling approaches to estimate the number of
$k$
-cliques in the
dense regions
of the graph. Specifically, we develop three novel dynamic programming based
$k$
-color set sampling techniques to efficiently estimate the
$k$
-clique counts, where a
$k$
-color set contains
$k$
nodes with
$k$
different colors. Since a
$k$
-color set is often a good approximation of a
$k$
-clique in the dense regions of a graph, our sampling-based solutions are extremely efficient and accurate. Moreover, the proposed sampling techniques are space efficient which use near-linear space w.r.t. graph size. We conduct extensive experiments to evaluate our algorithms using 8 real-life graphs. The results show that our best algorithm is at least one order of magnitude faster than the state-of-the-art sampling-based solutions (with the same relative error 0.1%) and can be up to three orders of magnitude faster than the state-of-the-art exact algorithm on large graphs.
